254 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF These equations give B , D These formulae can always be satisfied, and therefore it is always possible to represent the given group by a stream of common light combined with a stream of elliptically polarized light independent of the former. Moreover, there is only one way in which the group can be so represented. For, though the third of equations (19) gives two values for /3' complementary to each other, these values, as before explained, lead only to two ways of expressing the same result. If we choose that value of IB which is numerically the smaller, then among the different values of a', differing by 90°, which satisfy the fourth equation, we must choose one which gives to cos 2a' the same sign as C. 20. Let us now apply the principles and formulae which have just been established to a few examples. And first let us take one of the fundamental experiments by which MM. Arago and Fresnel established the laws of interference of polarized light, or [; * rather an analogous experiment mentioned by Sir John Herschel. ' The experiment selected is the following. Two neighbouring pencils of common light from the same source are made to form fringes of interference. A tourmaline, carefully worked to a uniform thickness, is cut in two, and its halves interposed in the way of the two streams respectively. It is found that when the planes of polarization of the two tourmalines are parallel the fringes are formed perfectly; but as one of the tourmalines is turned round in azimuth the fringes become fainter, and at last, when the planes of polarization become perpendicular to each other, the fringes disappear. Let the planes of polarization of the tourmalines be inclined at an angle a, and let it be required to investigate an expression for the intensity of the fringes. Since common light is equivalent to two independent streams, of equal intensity, polarized in opposite ways, let the original light be represented by two independent streams, having each an intensity equal to unity, polarized in planes respectively parallel and perpendicular to the plane of polarization of the first tourmaline. If c, c' be the co-